Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most $s$, we give an estimator using $O(s^2/\varepsilon^2)$ copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds $r,s$ as $\min\{r,s\}^2/\varepsilon^2$ up to logarithmic factors in the smaller rank. The same estimator gives the upper bound $O(r(\operatorname{tr}\sqrtσ)^2/\varepsilon^2)$ for a rank-$r$ reference $σ$. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra $λ_i\propto i^{-α}$ with fixed $1<α\le2$, as $\varepsilon\downarrow0$ with $r\ge C_α\varepsilon^{-2/(α-1)}$, the bounds determine the complexity as $\widetildeΘ_α(\varepsilon^{-4/(α-1)})$. In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.